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Fundam. Prikl. Mat., 2001, Volume 7, Issue 2, Pages 495–513 (Mi fpm571)  

This article is cited in 10 scientific papers (total in 10 papers)

Non-commutative Gröbner bases, coherentness of associative algebras, and divisibility in semigroups

D. I. Piontkovskii

M. V. Lomonosov Moscow State University

Abstract: In the paper we consider a class of associative algebras which are denoted by algebras with $R$-processing. This class includes free algebras, finitely-defined monomial algebras, and semigroup algebras for some monoids. A sufficient condition for $A$ to be an algebra with $R$-processing is formulated in terms of a special graph, which includes a part of information about overlaps between monomials forming the reduced Gröbner basis for a syzygy ideal of $A$ (for monoids, this graph includes the information about overlaps between right and left parts of suitable string-rewriting system). Every finitely generated right ideal in an algebra with $R$-processing has a finite Gröbner basis, and the right syzygy module of the ideal is finitely generated, i. e. every such algebra is coherent. In such algebras, there exist algorithms for computing a Gröbner basis for a right ideal, for the membership test for a right ideal, for zero-divisor test, and for solving systems of linear equations. In particular, in a monoid with $R$-processing there exist algorithms for word equivalence test and for left-divisor test as well.

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Bibliographic databases:
UDC: 512.552
Received: 01.12.1996

Citation: D. I. Piontkovskii, “Non-commutative Gröbner bases, coherentness of associative algebras, and divisibility in semigroups”, Fundam. Prikl. Mat., 7:2 (2001), 495–513

Citation in format AMSBIB
\by D.~I.~Piontkovskii
\paper Non-commutative Gr\"obner bases, coherentness of associative algebras, and divisibility in semigroups
\jour Fundam. Prikl. Mat.
\yr 2001
\vol 7
\issue 2
\pages 495--513

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    This publication is cited in the following articles:
    1. Tchoupaeva I., “Analysis of geometrical theorems in coordinate-free form by using anticommutative Grobner bases method”, Automated Deduction in Geometry, Lecture Notes in Artificial Intelligence, 2930, 2004, 178–193  mathscinet  zmath  isi
    2. S. A. Ilyasov, “Construction of the syzygy module in automaton monomial algebras”, J. Math. Sci., 142:2 (2007), 1933–1941  mathnet  crossref  mathscinet  zmath
    3. A. Ya. Belov, “Linear Recurrence Equations on a Tree”, Math. Notes, 78:5 (2005), 603–609  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. D. I. Piontkovskii, “Koszul Algebras and Their Ideals”, Funct. Anal. Appl., 39:2 (2005), 120–130  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Piontkovski, D, “Linear equations over noncommutative graded rings”, Journal of Algebra, 294:2 (2005), 346  crossref  mathscinet  zmath  isi
    6. Polishchuk, A, “Noncommutative Proj and coherent algebras”, Mathematical Research Letters, 12:1 (2005), 63  crossref  mathscinet  zmath  isi
    7. S. A. Ilyasov, “Recognition of Certain Properties of Automaton Algebras”, Journal of Mathematical Sciences, 152:1 (2008), 95–136  mathnet  crossref  mathscinet  zmath
    8. I. A. Ivanov-Pogodaev, “Finite Gröbner basis algebra with unsolvable problem of zero divisors”, J. Math. Sci., 152:2 (2008), 191–202  mathnet  crossref  mathscinet  zmath
    9. Piontkovski, D, “Koszul algebras associated to graphs”, International Mathematics Research Notices, 2006, 84040  mathscinet  zmath  isi  elib
    10. Piontkovski, D, “Coherent algebras and noncommutative projective lines”, Journal of Algebra, 319:8 (2008), 3280  crossref  mathscinet  zmath  isi
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